Big O and Order of growth

Question

Let’s say we have the following function:

$$T(n) = 33*n^2 + 55 - 12*n ∈ O(n^2)$$

Is it correct to say that $n^2$ is order of growth of $T(n)$?

And more commonly:

If $T(n) ∈ O(G(n))$

Is it correct to say that $G(n)$ is order of growth of $T(n)$?

And the last question. Let’s say we have the following function without any Big O specification:

$$T(n) = 2*n^3 + 55 + 12*log n$$

Is it correct to say that $n^3$ is order of growth of $T(n)$?

Answer 1

To anwser your question directly. When talking about Big-oh notation, you always take the biggest growth since big-oh is your upper limit so you are correct saying your function is n^3. Now when you say order of growth I assume you do not only mean upper limit, that's where you would use Big-Theta. When using big Theta you have to be able to set upper and lower limit to your function.